Prepare the dough: In a bowl, mix flour, sugar, baking powder, salt, and cinnamon. Add milk, butter, egg, and vanilla. Stir until smooth.
Roll out dough: On a floured surface, roll the dough to about 1/2-inch thickness.
Cut donuts: Use a donut cutter (or two different-sized cookie cutters) to cut the dough into donut shapes.
Heat oil: In a deep fryer or large pot, heat oil to 350°F (175°C).
Fry donuts: Fry donuts in batches for 1-2 minutes per side, until golden brown.
Make glaze: In a bowl, whisk together coffee, powdered sugar, and vanilla extract until smooth.
Glaze donuts: Dip warm donuts into the glaze, coating them thoroughly.
Cool: Let the donuts rest on a cooling rack for a few minutes before serving.